This might be better served in the chemistry section for reasons I'm about to explain.

So this property of evaporation vs boiling strikes a lot of people as being sort of odd, but the answer lies in the very nature of liquids. So, all (maybe most?) have a 'vapor pressure,' that is, there are some free molecules of the liquid that rest just above the liquid itself. Even mercury technically has a vapor pressure, albeit quite small.

Liquids, and some solids, exhibit vapor pressure because of chance and, more importantly, the energy distribution of a group of molecules at the same temperature. Not all the molecules in a collection are at the same energy level, and the usual entropy obeying model shows that most of the molecules of a collection are in the lowest state and some in the 'middle' state and just a few in the highest state (this is pretty simplistic but you get the idea).

These higher energy molecules can escape sometimes on the surface of the liquid as the weak bonds of the liquid can't hold every molecule in all the time. If you're in a closed environment, these molecules will escape into the area above the liquid until you reach a sort of equilibrium position, where the rate of chance escaping molecules equals the chance returning molecules that have drifted back down.

If you raise the temperature of the liquid, you get a higher energy distribution in which more molecules have the energy required to escape one by one, so the vapor pressure will rise. This is why being in a location that's warm and wet causes 'humidity.'

So evaporation can occur when the liquid is at a reasonable temperature and is exposed to an open environment, or one where the vapor molecules are easily swept away, think blowing a fan over warm water. This would cause the rate of returning molecules to drop, but the rate of exiting molecules would be mostly unchanged or rise due to the turbulence of the wind or whatever is going on in your environment.

Boiling occurs when, and this is quite cool, when the vapor pressure of a liquid has reached the ambient pressure (usually atmospheric pressure). So when you heat a liquid, the energy distribution rises and rises until most of the molecules are in a high enough state to exert an equal pressure on the atmosphere. Once this occurs, molecules will escape en masse.